Lumer–Phillips theorem

In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.

Contents

Statement of the theorem

Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup if and only if[1]

  1. D(A) is dense in X,
  2. A is closed,
  3. A is dissipative, and
  4. A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.

Variants of the theorem

Reflexive spaces

Let A be a linear operator defined on a linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if[2]

  1. A is dissipative, and
  2. A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.

Note that the conditions that D(A) is dense and that A is closed are dropped in comparison to the non-reflexive case. This is because in the reflexive case they follow from the other two conditions.

Dissipativity of the adjoint

Let A be a linear operator defined on a dense linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if[3]

In case that X is not reflexive, then this condition for A to generate a contraction semigroup is still sufficient, but not necessary.[4]

Quasicontraction semigroups

Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a quasi contraction semigroup if and only if

  1. D(A) is dense in X,
  2. A is closed,
  3. A is quasidissipative, i.e. there exists a ω ≥ 0 such that ωI − A is dissipative operator, and
  4. A − λ0I is surjective for some λ0 > ω, where I denotes the identity operator.

Examples

\langle u, A u \rangle = \int_0^1 u(x) u'(x) \, \mathrm{d} x = - \frac1{2} u(0)^2 \leq 0,
so that A is dissipative. The ordinary differential equation u' − λu = f, u(1) = 0 has a unique solution u in H1([0, 1]; R) for any f in L2([0, 1]; R), namely
u(x)={\rm e}^{\lambda x}\int_x^1 {\rm e}^{-\lambda t}f(t)\,dt
so that the surjectivity condition is satisfied. Hence, by the reflexive version of the Lumer–Phillips theorem A generates a contraction semigroup.

There are many more examples where a direct application of the Lumer–Phillips theorem gives the desired result.

In conjunction with translation, scaling and perturbation theory the Lumer–Phillips theorem is the main tool for showing that certain operators generate strongly continuous semigroups. The following is an example in point.

Notes

  1. ^ Engel and Nagel Theorem II.3.15, Arent et. al. Theorem 3.4.5, Staffans Theorem 3.4.8
  2. ^ Engel and Nagel Corollary II.3.20
  3. ^ Engel and Nagel Theorem II.3.17, Staffans Theorem 3.4.8
  4. ^ There do appear statements in the literature that claim equivalence also in the non-reflexive case (e.g. Luo, Guo, Morgul Corollary 2.28), but these are in error.
  5. ^ Engel and Nagel Exercise II.3.25 (ii)

References